Axiomatic Thinking II

Preface
Acknowledgements
Contents Overview of Vol. 1
Contents
Editors and Contributors
About the Editors
Contributors
Part I Logic I(2)
1 A Framework for Metamathematics
1.1 Hilbert's Program Revisited
1.2 Non-constructive Principles of Metamathematics
1.2.1 What We Need
1.2.2 What We Obtain
1.3 Conclusion
References
2 Simplified Cut Elimination for Kripke-Platek Set Theory
2.1 Introduction
2.2 Kripke-Platek Set Theory
2.3 A Tait-Style Reformulation of KP
2.4 An Ordinal System for the Bachmann-Howard Ordinal
2.5 Derivation Operators
2.6 The Infinitary Proof System IP
2.7 Partial Soundness and Completeness of IP
2.8 Embedding of KPT into IP
2.9 Predicative Cut Elimination
2.10 Collapsing Theorem
References
3 On the Performance of Axiom Systems
3.1 Introduction
3.2 Characteristic Ordinals
3.2.1 Semi-formal Systems
3.2.2 The Ordinals πmathfrakM and πmathfrakM(T)
3.2.3 Basics of Ordinal Arithmetic and Cut-Elimination
3.2.4 Boundedness
3.2.5 An Example
3.3 Analytical Universes Above mathfrakM
3.3.1 Spector Classes
3.3.2 Fixed-Point Theories
3.3.3 Collapsing
3.4 Ordinal Analysis for Arithmetical Universes
3.4.1 Upper Bounds
3.4.2 Lower Bounds
3.5 Provably Recursive Functions
3.6 Conclusion
References
4 Well-Ordering Principles in Proof Theory and Reverse Mathematics
4.1 Introduction
4.1.1 Reverse Mathematics
4.2 History
4.2.1 2mathfrakX and Arithmetical Comprehension
4.2.2 ACA0+ and εmathfrakX
4.2.3 Proof Idea of (1)(2) of Theorem 4.11
4.3 Towards Impredicative Theories
4.3.1 The Bachmann Revelation
4.3.2 Associating a Dilator with Bachmann
4.4 Towards a General Theory of Ordinal Representations
4.4.1 Feferman's Relative Categoricity
4.4.2 Girard's Dilators
4.5 Higher Order Well-Ordering Principles
4.5.1 Bachmann Meets a Dilator
4.5.2 Deduction Chains in PAmathfrakX
4.5.3 A Glimpse of Anton Freund's Work
4.6 There Are Much Stronger Constructions Than Bachmann's
References
Part II Mathematics II(2)
5 Reflections on the Axiomatic Approach to Continuity
References
6 Abstract Generality, Simplicity, Forgetting, and Discovery
6.1 An Articulating Generalization: Riemannian Manifolds
6.2 A Hypothetical Example of a Unifying Generalization
6.3 Generalization at the Origin of Abstract Algebra
6.4 Forgetting the Details, for a Time
6.5 Schemes
References
7 Varieties of Infiniteness in the Existence of Infinitely Many Primes
7.1 Introduction
7.2 The Axiom System and Some Basic Facts for the First Proof
7.3 Proof of the Infinity of Primes Based on the Co-Primeness of the Fermat Numbers
7.4 Euclid's Proof for the Cofinality of Primes
7.5 Comparing Notions of Infinity
References
8 Axiomatics as a Functional Strategy for Complex Proofs: The Case of Riemann Hypothesis
8.1 Axiomatics, Analogies, Conceptual Structures
8.2 Navigating Within the Mathematical Hymalayan Chain
8.3 Riemann's ζ-Function
8.3.1 The Distribution of Primes
8.3.2 Definitions of ζ(s)
8.3.3 Mellin Transform, Theta Function, and Functional Equation
8.3.4 Zeroes of ζ(s)
8.3.5 Riemann Hypothesis
8.3.6 The Problem of Localizing Zeroes
8.4 Explicit Formulas
8.4.1 Riemann's Explicit Formula
8.5 Local/Global in Arithmetics
8.5.1 Dedekind-Weber Analogy
8.5.2 Weil's Description of Dedekind-Weber Analogy
8.5.3 Valuations and Ultrametrics
8.5.4 p-Adic Numbers
8.5.5 Hensel's Geometric Analogy
8.5.6 Places
8.5.7 Local and Global Fields
8.6 The RH for Elliptic Curves Over mathbbFq (Hasse)
8.6.1 The ``Rosetta Stone''
8.6.2 The Hasse-Weil Function
8.6.3 Divisors and Classical Riemann-Roch (Curves)
8.6.4 Divisors and Classical Riemann-Roch (Surfaces)
8.6.5 RR for Curves Over mathbbFq
8.6.6 The Frobenius Morphism
8.6.7 RH for Elliptic Curves (Schmidt and Hasse)
8.7 Weil's ``Conceptual'' Proof of RH
8.8 Connes' Strategy: ``A Universal Object for the Localization of L Functions''
8.8.1 Come Back to Arithmetics
8.8.2 The Hasse-Weil Function in Characteristic 1: Soulé's Work
8.8.3 Semi-rings and Semi-fields of Characteristic 1
8.8.4 The Arithmetic Topos mathfrakA=( mathbbNtimes"0362mathbbNtimes,mathbbZmax)
8.9 Conclusion
References
Part III Other Sciences III(2)
9 What is the Church-Turing Thesis?
9.1 Introduction
9.2 What Is Computed?
9.3 How Is It Computed?
9.4 What Can Be Proved?
9.4.1 Against Provability
9.4.2 In Favour of Provability
9.4.3 What Does It Mean to Disprove the Thesis?
9.5 The Mathematical Thesis
9.5.1 Non-empirical Arguments in Favour of the Thesis
9.5.2 Disproving the Thesis
9.6 The Physical Thesis
9.6.1 Proving the Thesis
9.6.2 Disproving the Thesis
9.7 Conclusion
References
10 Axiomatic Thinking in Physics—Essence or Useless Ornament?
10.1 Prologue
10.2 Introduction
10.3 Some Examples
10.3.1 Isaac Newton and Mechanics
10.3.2 Heinrich Hertz and Modern Analytical Mechanics
10.3.3 Constantin Carathéodory and Classical Thermodynamics
10.3.4 Max Born and the ``Old'' Quantum Mechanics
10.3.5 Werner Heisenberg and Quantum Field Theory
10.4 Space-Time
10.4.1 Minkowski Space
10.4.2 General Relativity
10.5 Conclusions and Summary
References
11 Axiomatic Thinking—Applied to Religion
11.1 On the Possibility of Applying Axiomatic Thinking to Religion
11.1.1 Applying Logic to Religion
11.1.2 Religious Discourse
11.1.3 Religious Texts. Example: The Bible of the Christian Religion
11.1.4 The Two Kinds of Belief
11.2 Applying Axiomatic Thinking to Religion: Logical Language
11.2.1 Logic
11.2.2 Set-Theoretic Elementhood: ε
11.2.3 Operators
11.2.4 Individual Constant
11.2.5 Quantifiers
11.2.6 Modal Operators
11.2.7 Interpretation
11.3 Applying Axiomatic Thinking to Religion: Omniscience and Omnipotence
11.3.1 Omniscience
11.3.2 Problems Concerning Theorem T6: Necessary Knowledge
11.3.3 Omnipotence
References